An inequality for the norm of a polynomial factor

TL;DR

This paper establishes a sharp inequality relating the norms of a monic polynomial and its factor on a compact set, using potential theory to determine the optimal constant, with applications to specific geometric sets.

Contribution

The paper introduces an asymptotically sharp inequality for polynomial factors' norms and determines the best constant via potential theoretic methods, extending to specific sets like disks and segments.

Findings

Derived a sharp inequality for polynomial factors' norms.

Identified the optimal constant using potential theory.

Applied results to disks and segments.

Abstract

Let $p(z)$ be a monic polynomial of degree $n$, with complex coefficients, and let $q(z)$ be its monic factor. We prove an asymptotically sharp inequality of the form $\|q\|_{E} \le C^n \|p\|_E$, where $\|\cdot\|_E$ denotes the sup norm on a compact set $E$ in the plane. The best constant $C_E$ in this inequality is found by potential theoretic methods. We also consider applications of the general result to the cases of a disk and a segment.